Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions

The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series; 2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians; 3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation; 4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.

n n The first case is elementary whereas, in treating the second case, we give non-trivial extensions of one known result for power series. §3 contains results concerning the type of asymptotic variation of a Wronskian whose arguments are functions with a definite index of asymptotic variation at +∞ so completing the extensive study of the asymptotic behaviors of Wronskians developed in two previous papers. The obtained results are quite natural and are based on the asymptotic study of a Vandermonde determinant with a gap in the exponents, a study which parallels the analogous investigation for standard Vandermondians in the previous papers. §4 contains a comparison between the two main standard approaches to the concept of "type of asymptotic variation": via an asymptotic differential equation or an asymptotic functional equation. The theory developed in [1] [2] is necessarily based on asymptotic differential equations in order to define higher-order types of variation for differentiable functions, whereas the more general Karamata theory is based on asymptotic functional equations. We show that for a function with a monotonic derivative the two approaches coincide for each one of the studied classes of functions, a result already known for regular variation. §5 contains a discussion about the concept of logarithmic variation starting from suitable asymptotic functional equations and showing, as in the previous section, the equivalence with corresponding asymptotic differential equations. Some of the results may be found in the literature but hidden in a quite-complicated general theory. The studied concept (namely, three related concepts) completes the list of the fundamental types of asymptotic variation in the way that we wished to systematize this theory. §6 gives results on the inverse of a function with a definite type of exponential variation. The results require some calculations and are clarified by the concepts of logarithmic variation. §7 contains some minor complements to the theory. §8 contains the conclusions about the whole theory developed in three papers. §9 contains a few bibliographical notes and a list of corrections for [1] [2].
The differentiation operators: ; : The logarithmic derivative: : Hardy's notations: The relation of "asymptotic similarity", " ( ) ( ) The relation of "asymptotic equivalence": The relation: lim ; x and a similar definition for notation where k α is termed the "k-th falling ( ≡ decreasing) factorial power of α ".
Everywhere the symbol " log x " stands for " ( ) e log x " := "the natural logarithm" of x.
Notation for the iterated natural logarithm: For the reader's convenience we give a list of the special classes of functions characterized in [1] [2] mentioning only the main facts to be used in the present paper.
Classes of functions and their main characterizations.
(I) (Index of asymptotic variation). If [ ) , f AC T ∈ +∞ , f ultimately > 0, its index of asymptotic variation at +∞ is defined as the value of the following  1 , , 1 n f AC T n − ∈ +∞ ≥ is termed "regularly varying at +∞ (in the strong sense) of order n" if each of the functions ( ) 1 , , , n f f f − ′  never vanishes on a neighborhood of +∞ and is regularly varying at +∞ with its own index of variation. If this is the case we use notation where 1 α is the index of f ′ and the index of ( ) Notice that the last derivative involved in ( 10) the reason of the last strict inclusion being that some derivatives of a smoothly-varying function may vanish or change sign infinitely often. The following sets of asymptotic relations, for a fixed α ∈  , are equivalent to each other:  0  large enough;  ,  ; 1 , ; (1.13a) or, equivalently, if: ; (Rapid variation of higher order). A function [ ) , n f AC T ∈ +∞ is called "rapidly varying at +∞ of order 2 n ≥ (in the strong restricted sense)" if all the functions are rapidly varying at +∞ in the above-specified sense and this amounts to say that the following conditions hold true as If f is rapidly varying at +∞ of order 2 n ≥ in the previous sense then all the functions to denote that f enjoys the properties in (1.14)-(1.15)-(1.16) plus the corresponding value ±∞ of the limit in (1.6 , i.e. , , 1 ; It follows that even ( ) ( ) wherein the correct index " +∞ " or " −∞ " is determined by the single limit For 0 c = there is no sign-restriction on the highest-order derivative ( ) turns out to be ultimately of one strict sign. More Notice that in our definition of higher-order variation f is allowed to be either >0 or <0, the essential point being that it ultimately assumes only one strict sign.
The reader must remember that this is a semiexpository paper like [1] [2] and, as such, some elementary or known facts are explicitly reported or proved to have an exposition self-contained and easily-read.

Types of Asymptotic Variation of Infinite Series
In [1] and [2] there are some results about the index of variation of a linear combination of functions belonging to one of the previously-studied classes; in this section we give some results about the type of asymptotic variation of an infinite series of such functions. We know from ( , there must be an essential difference between the two circumstances The simplest case is ( are absolutely and uniformly convergent on each bounded interval of [ ) (II) Assume all the conditions in part (I) with the exception that (2.5) is now replaced by (2.9) or, more generally, by Then all the conclusions in part (I) still hold true. In the special case ( ) ( ) is an asymptotic scale as well provided that " 0 n n α ≠ ∀ " because of relations " ( ) ( ) Remarks. Conditions in (2.4), (2.5), (2.10) are a kind of uniformity respectively for the infinite families of asymptotic relations: These conditions cannot be dispensed with and even in a simple case such as do not in themselves grant (2.10) as shown by the counterexample of For part (II) instead of (2.15) we now have: ; n n n n n n n n n k n k n k n n n n n k for 2 k ≥ and the subsequent conclusions are still valid.  An elementary example. For any function φ such that: we have the estimates: And if 0 1 M < < and k ∈  : In both cases " ( ) ( ) ( ) ( ) the circumstance " 0 α > " being inconsistent with condition " A less elementary example. Consider the sequence of "modified iterated logarithms": Let us now examine the case (2.3), the classical case being that of a power series with an infinite radius of convergence; here coefficients of nonconstant signs may generate entire functions with no definite type of asymptotic variation at +∞ such as the trigonometric functions, hence in this case we must restrict our study to positive coefficients. A problem solved in American Mathematical Monthly, [3], states that if the function   ; . n n n n n n n n n n n n n n Thesis: Remark. Of course, instead of (2.30) it is enough to assume that Proof. From (2.31)-(2.32) we get (2.34) as: . n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n Case " 0 < α ≤ +∞ ", which implies " ( ) 0 n x x n φ′ > ∀ ∀ ". First step: x Second step. Using a different device we get for each fixed k:  x and (2.38) yields: for each fixed k. (2.42) Taking the limit as k → ∞ : and this last limit implies the limit " +∞ " for the inverted ratio.
Recall: 0 n α < and nondecreasing. First step: Second step. For each fixed k: whereas for the second ratio, as it stands without the absolute value, we have: ≥ for each fixed k; (2.49) and taking the limit as k → ∞ : which, together with (2.46) yields " Consistently with the statement of Proposition 2.2 the circumstance " 0 n k α − =" for some value of k and at most one value of n, say n p = , is treated by splitting the series Second case. If lim n n α = +∞ then: In fact, assuming 0 n n α > ∀ and with " [ ] n α = integer part of n α ", we have where the second series on the right is a power series majorized by the convergent series 2) For The cases " constant n α = ∈  " deserve a separate brief discussion as they can be treated quite simply. The following results do not require any growth-order chain between the n φ 's. (2.57) Then the function ( ) ( ) : , Without either the restriction on the signs of n c or the conditions in (2.58) no definite conclusion on the type of variation of f can be drawn.
Proof. Notice that (2.56) implies ( ) 0 n L x > for x large enough and that we may always assume 0 T > . For 0 α = we have: , ; , ; whence it follows that ( ) ( ) Counterexamples. Let L be any function such that:    x Proof. Both proofs are trivial using the stated assumptions of "uniformity with respect to n" of the relations " and analogously for n α = −∞ .
 For series, with positive coefficients, of functions having hypo-exponential, or exponential, or hyper-exponential variation at +∞ corresponding results hold true suppressing the factor x on the left of the asymptotic relation in (2.31). All provided that the remaining assumptions in Proposition 2.2 are satisfied.
Three more examples.
The following examples are not included in the previous results but elucidate techniques based on the type of asymptotic variation.
1) Consider the sequence: This is a sequence of slowly varying functions which is locally uniformly convergent to a regularly-varying function of index 1. The factor ( ) n r x satisfies: . But direct calculations yield more precise information: valid for e x ≥ ; whence, integrating the ratio f f ′ : The relations in the first line of (2.68) imply the two limits If the two conditions in (2.30) hold true then: ≥ " yields the global estimates: and obviously: 3) For a bounded sequence of real numbers of arbitrary signs, n c M ≤ , consider the following two series: We also have the estimates: hence f is positive, strictly decreasing and:

Type of Asymptotic Variation of a Wronskian
In two papers [5] [6] the author described a number of techniques to obtain the asymptotic behaviors of Wronskians whose entries are regularly-or rapid- is an ordered n-tuple of complex numbers with 2 n ≥ , its Vandermondian with a one-unit gap in the highest exponent is defined as the number: denotes the standard Vandermondian. (II) The following formula holds true: which means that each i φ is smoothly varying of order n and index i a according to Definition 3.2 in ([1]; p. 803). Then: i.e. W is regularly varying of order n and index " ( ) Applying the procedure in (3.4)-(3.5) it is seen that the last determinant equals , and by Lemma 3.1 we get: whence the second relation in (3.7) follows.  To obtain a result in the case of rapid variation we need a correct statement of the analogue of Theorem 6 in ( [5]; pp. 11-12) for the determinant V  . Rereading the proof of this theorem we get the following claims. Lemma 3.3. Let 1 , , n g g  and 1 , , n f f  be functions defined on a deleted neighborhood of 0 x ∈  and such that: The following relations hold true. (I) The general estimate n n n n p where ′  denotes the set of all permutations of the n-tuple ( ) Formula (3.11) must be read with the agreement that " ( ) , .
[where " f g  " means " ( ) then relation (3.11) takes the simpler form: noticing, in the right-hand side, the lack of n f which is one of the functions i f with the lowest growth-order. The two most meaningful cases are highlighted in the following statements.
If all the i f 's have the same growth-order in the sense that for a fixed f and arbitrary pairwise-distinct constants i c , then: In the special case ( ) 1 f x ≡ we get: Hints for the proof. When rereading Theorem 6 in ( [5]; pp. 11-14) the reader will notice that the agreement " ( ) is not explicitly stated in the statement but it is clearly specified in the proof ( [5]; p. 13, line 8 from below). At this point of the proof in [5] the estimate in (3.11) is proved; to proceed, the reader will replace the quantity in formula (95) in ( [5]; p. 13) with: which, in particular, are satisfied by functions which are rapidly varying at +∞ of order 1 n − in the strong restricted sense of ( which means that W has the same type of asymptotic variation as 1 φ according to the concept first formulated by Hardy [8] and, in particular, W has the same type of exponential variation as 1 φ according to ([2]; Def. 8.1, p. 832).
for some fixed function φ and pairwise-distinct constants i c then, as x → +∞ : and under conditions in (3.25):

Asymptotic Differential Equations versus Asymptotic Functional Equations
The theory developed in [1] [2] concerns functions differentiable a certain number of times and it starts from various types of asymptotic relations; for instance, the relation defines the basic concept of regular variation of index α in the strong sense, a concept (but not the locution) dating back to Hardy [8]. The above relation may be termed an "asymptotic differential equation". But, as mentioned in ( It is known that for a function f with a monotonic derivative conditions (4.1a) and (4.1b) are equivalent and we show in this section that the same is true for the pair of equations (differential and functional) pertinent to each of the other types of variation: rapid, hypo-exponential, exponential or hyper-exponential. The first result of this type has been proved by Lamperti ([9]; pp. 382-383) for everywhere-differentiable functions using the Lagrange mean-value formula: and where the role of φ may be played indifferently by the left or the right derivative of f , both existing everywhere and coinciding except possibly on a countable set: φ non-decreasing for convexity and non-increasing for concavity. If " f ′ exists on I save possibly a set N  of Lebesgue measure zero and is monotonic on \ I N  and if the absolute continuity of f is explicitly assumed" then for any three numbers in I, x z y < < , and, say, f ′ non-decreasing we have the inequalities: whence, by standard calculations, we get the inequality: In fact: In the setting of absolute continuity, condition " f ′ monotonic" even in its weakest meaning unambiguously refers to a function whose left and right derivatives exist as finite numbers at each interior point, coincide except possibly on a countable set and are monotonic with the appropriate type of monotonicity.
Moreover, an asymptotic relation involving f ′ such as, for instance (4.1a) may be read in any of the following three ways: whereas, strictly speaking, the shortened notation in (4.1a) refers to the case of an everywhere-differentiable function. The following two theorems show that the differential and the functional approaches to the theory of asymptotic variation coincide for functions which are ultimately either concave or convex which is certainly the case of functions whose first derivatives are regularly varying of index 0 ≠ or rapidly varying. In the proofs use is made of the integral representation in (4.2). ; ; , for each fixed and 0 1 , ;  (4.12) Proof of Theorem 4.1. All the inferences from the right to the left in both theorems elementarily follow from the integral representations and are to be found in ( [1]; §5) and in ( [2]; §8). Here we have to prove the converses. All the proofs are based on estimating the integral in terms of f ′ where f ′ indifferently stands for the right or left derivative of f which both exist everywhere. Proof of (4.6); for f ′ non-decreasing we have: whence: and letting x → +∞ : and we get our claim letting 1 λ + → . For f ′ non-increasing use the same argument reversing the inequalities in (4.13). For the proof of (4.7) we have: The result in (4.8) is brought back to (4.7) putting ( ) ( )   ( )   1  for  0,  1  for  ,  for  0,  for  ,  lim  lim  0  for 0  1,  0  for  0,  for  1,  for  0; But a correspondence between the monotonicity of f ′ and g′ is not granted; hence it is better to give direct proofs of Theorem 4.2 following the same patterns as above. We write them down only for " f ′ non-decreasing".
For the proof of (4.9) notice that the claim does not depend on the sign of f ; hence we may suppose " 0 f ′ ≥ " changing, if necessary, the sign of f . For " f ′ non-decreasing" we have: For the proof of (4.10) and " f ′ non-decreasing" we have: whence: and as x → +∞ : And for the proof of (4.11) we write: and the assertion follows by taking first the "lim inf" as x → +∞ and then the limit as 0 Remarks about growth-estimates. 1) One of the basic properties of the functions in the studied classes is the estimates, though rough, of their growth-orders.
For the subclasses defined via asymptotic differential equations the estimates are elementarily inferred from the pertinent integral representations whereas for the general classes it is true that they are inferred from suitable integral representations as well, but such representations are not elementary facts but consequences of the nontrivial core of the theory, namely the "uniform convergence theorems". , T +∞ the two functional equations in (4.7) imply that f is non-decreasing and admits of the integral representation: where the measurable functions , , z η ξ are such that: z is non-decreasing, x ξ → +∞ as x → +∞ (This is the case of Theorem 4.1 wherein the monotonicity of f ′ implies the ultimate monotonicity of f ). From (4.26) the following estimate is trivially inferred: By putting ( ) ( ) : x g x f e = one gets the corresponding growth-estimates for exponential variations listed in ( [2]; §8).
2) In the case of (4.7) the contingency " f ′ non-increasing" is excluded otherwise we would have: contradicting either the growth-estimate in (4.27b) or the positivity of f .
3) In passing we point out that the above-mentioned "uniform convergence theorems" imply that a function satisfying one of the asymptotic functional equation mentioned in Theorems 4.1-4.2 satisfies a quite stronger functional equation. A list appears in ( [1]; §5) inferred from the simple integral representations valid when the corresponding asymptotic differential equations hold true: the pertinent calculations are elementary, though not trivial, implicitly using the uniform convergence with respect to the parameter. In the more general context we have: , for each 0, measurable, without any monotonicity restriction, , , ; a result inferred from the factorization

Concepts Related to Logarithmic Variation
The concept of regular variation of order 0 α ≠ generalizes the asymptotic behavior of a power whereas the three concepts of exponential variation generalize the asymptotic behaviors of the exponential of a power. These generalizations are quite natural whereas the concept of slow variation is not the appropriate generalization of the behavior of the logarithm: it shares some asymptotic properties of the logarithm but it encompasses functions with orders of growth either greater than the order of each positive power of the logarithm or less than the order of each negative power of the logarithm such as the functions: Looking at the asymptotic functional equation of a slowly-varying function  , we see no link with the parameter λ , whereas the arithmetic functional equation characterizing the logarithm, if interpreted asympotically, gives an expression for the remainder. One of the possible asymptotic counterparts of (5.1b), as x → +∞ , is the asymptotic functional equation: It is known, (  obviously the behaviors of such functions have nothing to partake of the intuitive meanings associated with "logarithmic variation". For this reason someone might prefer to use the locutions of "logarithmically varying" and "hypo-logarithmically varying" exclusively for functions which, besides satisfying the appropriate above-specified equation, enjoy additional properties such as: strict positivity, or monotonicity, or divergence to +∞ as x → +∞ , or the last two properties. But this is a matter of agreement. Now we collect the essential properties of the three classes using the shortened locution " f differentiable" to mean that: with the agreement that an asymptotic relation involving f ′ , say for an absolutely continuous f is to be meant as " where N is a suitable set of measure zero. Three admissible meanings of the locution " f ′ monotonic" have been highlighted at the outset of the preceding section and the proofs involving this property may be done using the integral mean-value theorem which applies to each of the three cases. 2) f satisfies the estimate: Remarks. 1) Some people would like to add to the above definition a condition such as "strict positivity or monotonicity or divergence" for the sole function f (and not for its derivatives!) to adhere more consistently to the intuitive notion of logarithmic variation; but this is a matter of agreement as noticed above.
2) The asymptotic estimate (5.9) obviously implies (5.8) but the converse fails regardless of any monotonicity restriction; in fact for, say, a 1 C function we have: And in the case " ( ) ( )  ; Some examples. 1) The standard ones. The following functions are hypo-logarithmically varying at +∞ of a non-trivial type: The first of the foregoing function shows the obvious fact that no one of the three classes in Definition 5.1 is closed under multiplication.
The following examples concern various types of compositions.
2) Logarithm of a regularly-varying function: log logarithmically varying at , as log log 3) A regularly-varying function of the logarithm. The general result is: ( ) quasi-logarithmically varying at , , asymptotically sublinear at , hypo-logarithmically varying at .
In fact the assumptions mean that g satisfies " ( ) ( ) ( ) The most meaningful contingency for f in (5.27) is: in particular : , 1, so generalizing the first example in 5.24 .
The case " ( ) 1 f ∈ +∞  " needs restrictions: ( ) hypo-logarithmically varying at , , asymptotically uniformly continuous at , hypo-logarithmically varying at ; whose proof is as above, noticing that now " ( ) ( ) ( ) hence  is hypo-logarithmically varying at +∞ of order n according to our , then: and it is hypo-logarithmically varying at +∞ of order n.
which imply: and from these we get: x For any higher derivative we use Faà Di Bruno's formula ( [2]; formulas (6.1)-(6.2), p.818): where the summation is taken over all possible ordered k-tuples of non-negative integers j i such that ( ) In conclusion, the discussion in this section gives a theoretical description of the possible concepts related to a "logarithmic variation" at +∞ but these seem to have a limited import for practical applications, though linked with the general theory of asymptotic variation. An application is presented in the next section.

Inverse of a Function with a Type of Exponential Variation
This is termed "Open Problem 4" in ( [2]; p. 866); a complete treatment requires some calculations and it is appropriately understood in the context of logarithmic variation. Let f have a definite type of exponential variation namely, either: The restriction in (6.2) is unavoidable as the class ( ) 0 +∞  contains functions with different types of asymptotic variation whose inverses (if they exist) may have quite different behaviors and no definite general result may be given. Assumptions in (6.1) or (6.2) imply that f is ultimately positive and f ′ has ultimately one strict sign, hence f has an inverse and analogously we get that in each case: because the following sum is zero: ( )      We hope that our exposition (in these three semi-expository papers) of the general theory of "higher-order types of asymptotic variation" will reveal exhaustive and apt to applications. Until now the author has applied this theory to obtain non-elementary results about the asymptotic behaviors of Wronskians [5] [6]. A minor fact remain unsolved and this is "Open Problems 1 and 3" in ( [2]; p. 866) and an open problem stated after relation (2.110) in ( [1]; p. 796). We think that the three problems are interrelated and if one finds out a counterexample for one of them, then suitable adaptations will yield counterexamples for the other questions. This puzzled the author and a lot of scribbled-out sheets of paper have been wasted.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.